### Abstract

In this paper, we introduce a cofibrant simplicial category that we call the free homotopy coherent adjunction and characterise its n-arrows using a graphical calculus that we develop here. The hom-spaces are appropriately fibrant, indeed are nerves of categories, which indicates that all of the expected coherence equations in each dimension are present. To justify our terminology, we prove that any adjunction of quasi-categories extends to a homotopy coherent adjunction and furthermore that these extensions are homotopically unique in the sense that the relevant spaces of extensions are contractible Kan complexes. We extract several simplicial functors from the free homotopy coherent adjunction and show that quasi-categories are closed under weighted limits with these weights. These weighted limits are used to define the homotopy coherent monadic adjunction associated to a homotopy coherent monad. We show that each vertex in the quasi-category of algebras for a homotopy coherent monad is a codescent object of a canonical diagram of free algebras. To conclude, we prove the quasi-categorical monadicity theorem, describing conditions under which the canonical comparison functor from a homotopy coherent adjunction to the associated monadic adjunction is an equivalence of quasi-categories. Our proofs reveal that a mild variant of Beck's argument is "all in the weights"-much of it independent of the quasi-categorical context.

Language | English |
---|---|

Pages | 802-888 |

Number of pages | 87 |

Journal | Advances in Mathematics |

Volume | 286 |

DOIs | |

Publication status | Published - 2 Jan 2016 |

### Fingerprint

### Keywords

- Adjunction
- Homotopy coherence
- Monad
- Monadicity
- Quasi-categories

### Cite this

*Advances in Mathematics*,

*286*, 802-888. https://doi.org/10.1016/j.aim.2015.09.011

}

*Advances in Mathematics*, vol. 286, pp. 802-888. https://doi.org/10.1016/j.aim.2015.09.011

**Homotopy coherent adjunctions and the formal theory of monads.** / Riehl, Emily; Verity, Dominic.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Homotopy coherent adjunctions and the formal theory of monads

AU - Riehl, Emily

AU - Verity, Dominic

PY - 2016/1/2

Y1 - 2016/1/2

N2 - In this paper, we introduce a cofibrant simplicial category that we call the free homotopy coherent adjunction and characterise its n-arrows using a graphical calculus that we develop here. The hom-spaces are appropriately fibrant, indeed are nerves of categories, which indicates that all of the expected coherence equations in each dimension are present. To justify our terminology, we prove that any adjunction of quasi-categories extends to a homotopy coherent adjunction and furthermore that these extensions are homotopically unique in the sense that the relevant spaces of extensions are contractible Kan complexes. We extract several simplicial functors from the free homotopy coherent adjunction and show that quasi-categories are closed under weighted limits with these weights. These weighted limits are used to define the homotopy coherent monadic adjunction associated to a homotopy coherent monad. We show that each vertex in the quasi-category of algebras for a homotopy coherent monad is a codescent object of a canonical diagram of free algebras. To conclude, we prove the quasi-categorical monadicity theorem, describing conditions under which the canonical comparison functor from a homotopy coherent adjunction to the associated monadic adjunction is an equivalence of quasi-categories. Our proofs reveal that a mild variant of Beck's argument is "all in the weights"-much of it independent of the quasi-categorical context.

AB - In this paper, we introduce a cofibrant simplicial category that we call the free homotopy coherent adjunction and characterise its n-arrows using a graphical calculus that we develop here. The hom-spaces are appropriately fibrant, indeed are nerves of categories, which indicates that all of the expected coherence equations in each dimension are present. To justify our terminology, we prove that any adjunction of quasi-categories extends to a homotopy coherent adjunction and furthermore that these extensions are homotopically unique in the sense that the relevant spaces of extensions are contractible Kan complexes. We extract several simplicial functors from the free homotopy coherent adjunction and show that quasi-categories are closed under weighted limits with these weights. These weighted limits are used to define the homotopy coherent monadic adjunction associated to a homotopy coherent monad. We show that each vertex in the quasi-category of algebras for a homotopy coherent monad is a codescent object of a canonical diagram of free algebras. To conclude, we prove the quasi-categorical monadicity theorem, describing conditions under which the canonical comparison functor from a homotopy coherent adjunction to the associated monadic adjunction is an equivalence of quasi-categories. Our proofs reveal that a mild variant of Beck's argument is "all in the weights"-much of it independent of the quasi-categorical context.

KW - Adjunction

KW - Homotopy coherence

KW - Monad

KW - Monadicity

KW - Quasi-categories

UR - http://www.scopus.com/inward/record.url?scp=84943802355&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2015.09.011

DO - 10.1016/j.aim.2015.09.011

M3 - Article

VL - 286

SP - 802

EP - 888

JO - Advances in Mathematics

T2 - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -